Nonlinear viscosity and velocity distribution function in a simple longitudinal flow

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A compressible flow characterized by a velocity field uₓ(x,t)=ax/(1+at) is analyzed by means of the Boltzmann equation and the Bhatnagar-Gross-Krook kinetic model. The sign of the control parameter (the longitudinal deformation rate a) distinguishes between an expansion (a>0) and a condensation (a<0) phenomenon. The temperature is a decreasing function of time in the former case, while it is an increasing function in the latter. The non-Newtonian behavior of the gas is described by a dimensionless nonlinear viscosity η*(a*), that depends on the dimensionless longitudinal rate a*. The Chapman-Enskog expansion of η* in powers of a* is seen to be only asymptotic (except in the case of Maxwell molecules). The velocity distribution function is also studied. At any value of a*, it exhibits an algebraic high-velocity tail that is responsible for the divergence of velocity moments. For sufficiently negative a*, moments of degree four and higher may diverge, while for positive a* the divergence occurs in moments of degree equal to or larger than eight.